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Popular Trigonometry >

sqrt(3)cos(x)+sin(x)<0

Solution

3 cos (x) + sin (x) < 0

Solution

- \frac{4 π}{3} + 2 πn < x < - \frac{π}{3} + 2 πn

Interval Notation

(- \frac{4 π}{3} + 2 πn, - \frac{π}{3} + 2 πn)

Decimal

- 4.18879 \dots + 2 πn < x < - 1.04719 \dots + 2 πn

Solution steps

3 cos (x) + sin (x) < 0

Rewrite using trig identities

Divide both sides by

2

\frac{3 cos ( x ) + sin ( x )}{2} < \frac{0}{2}

\frac{3 cos ( x ) + sin ( x )}{2} < \frac{0}{2} : \frac{3}{2} cos (x) + \frac{1}{2} sin (x) < 0

\frac{3 cos ( x ) + sin ( x )}{2} < \frac{0}{2}

Expand

\frac{3 cos ( x ) + sin ( x )}{2} : \frac{3}{2} cos (x) + \frac{1}{2} sin (x)

\frac{3 cos ( x ) + sin ( x )}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{3 cos ( x ) + sin ( x )}{2} = \frac{3 cos ( x )}{2} + \frac{sin ( x )}{2}

= \frac{3 cos ( x )}{2} + \frac{sin ( x )}{2}

= \frac{3}{2} cos (x) + \frac{1}{2} sin (x)

Expand

\frac{0}{2} : 0

\frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

\frac{3}{2} cos (x) + \frac{1}{2} sin (x) < 0

\frac{3}{2} cos (x) + \frac{1}{2} sin (x) < 0

\frac{3}{2} = sin (\frac{π}{3})

sin (\frac{π}{3}) cos (x) + \frac{1}{2} sin (x) < 0

\frac{1}{2} = cos (\frac{π}{3})

sin (\frac{π}{3}) cos (x) + cos (\frac{π}{3}) sin (x) < 0

Use the following identity:

cos (s) sin (t) + cos (t) sin (s) = sin (s + t)

sin (\frac{π}{3} + x) < 0

sin (\frac{π}{3} + x) < 0

For

sin (x) < a

, if

- 1 < a \leq 1

then

- π - arcsin (a) + 2 πn < x < arcsin (a) + 2 πn

- π - arcsin (0) + 2 πn < (\frac{π}{3} + x) < arcsin (0) + 2 πn

a < u < b

then

a < u and u < b

- π - arcsin (0) + 2 πn < \frac{π}{3} + x and \frac{π}{3} + x < arcsin (0) + 2 πn

- π - arcsin (0) + 2 πn < \frac{π}{3} + x : x > - \frac{4 π}{3} + 2 πn

- π - arcsin (0) + 2 πn < \frac{π}{3} + x

Switch sides

\frac{π}{3} + x > - π - arcsin (0) + 2 πn

Simplify

- π - arcsin (0) + 2 πn : 2 πn - π

- π - arcsin (0) + 2 πn

Use the following trivial identity:

arcsin (0) = 0

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= - π - 0 + 2 πn

- π - 0 + 2 πn = - π + 2 πn

= 2 πn - π

\frac{π}{3} + x > 2 πn - π

Move

\frac{π}{3}

to the right side

\frac{π}{3} + x > 2 πn - π

Subtract

\frac{π}{3}

from both sides

\frac{π}{3} + x - \frac{π}{3} > 2 πn - π - \frac{π}{3}

Simplify

x > 2 πn - π - \frac{π}{3}

x > 2 πn - π - \frac{π}{3}

Simplify

- π - \frac{π}{3} : - \frac{4 π}{3}

- π - \frac{π}{3}

Convert element to fraction:

π = \frac{π 3}{3}

= - \frac{π 3}{3} - \frac{π}{3}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- π 3 - π}{3}

Add similar elements:

- 3 π - π = - 4 π

= \frac{- 4 π}{3}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4 π}{3}

x > - \frac{4 π}{3} + 2 πn

\frac{π}{3} + x < arcsin (0) + 2 πn : x < 2 πn - \frac{π}{3}

\frac{π}{3} + x < arcsin (0) + 2 πn

Simplify

arcsin (0) + 2 πn : 2 πn

arcsin (0) + 2 πn

Use the following trivial identity:

arcsin (0) = 0

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= 0 + 2 πn

0 + 2 πn = 2 πn

= 2 πn

\frac{π}{3} + x < 2 πn

Move

\frac{π}{3}

to the right side

\frac{π}{3} + x < 2 πn

Subtract

\frac{π}{3}

from both sides

\frac{π}{3} + x - \frac{π}{3} < 2 πn - \frac{π}{3}

Simplify

x < 2 πn - \frac{π}{3}

x < 2 πn - \frac{π}{3}

Combine the intervals

x > - \frac{4 π}{3} + 2 πn and x < 2 πn - \frac{π}{3}

Merge Overlapping Intervals

- \frac{4 π}{3} + 2 πn < x < - \frac{π}{3} + 2 πn

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